Error, counting

Flow cytometer cell counts are much more precise and more accurate than hemocytometer counts. Hemocytometer cell counts are subject both to distributional (13) and sampling (14—16) errors. The distribution of cells across the surface of a hemocytometer is sensitive to the technique used to charge the hemocytometer, and nonuniform cell distribution causes counting errors. In contrast, flow cytometer counts are free of distributional errors. Statistically, count precision improves as the square root of the number of cells counted increases. Flow cytometer counts usually involve 100 times as many cells per sample as hemocytometer counts. Therefore, flow cytometry sampling imprecision is one-tenth that of hemocytometry. 

To the analytical chemist, a standard deviation31 is a logical figure of merit for the comparison of detectors. We shall merely introduce the important subject of counting errors (10.2) here. For present purposes, it suffices to know that these errors are predictable, and that they set a lower limit to the standard deviation in an analytical method that depends upon measuring the intensity of an x-ray beam by an integrating detector. 

In Table 9-2, the deviations for magnesia and alumina are comparable to those expected from the counting errors (10.2) in the other cases, the deviations are larger All these results must be regarded as highly promising fQr a production spectrograph on a difficult problem ... 

Under the simplest conditions, the standard counting error is approximately equal to the square root of the total number of counts or... 

The proof of this relationship is one of the triumphs of probability theory. The underlying considerations will be outlined here because they are basic to an understanding of the counting error. These considerations are most obviously applicable to radioactive systems, and it was to these that they were first applied.3... 

Comparison of Equation 10-7 with Equation 10-1 results in two important conclusions, of which the second is the less obvious. First, if one equation represents a Gaussian distribution, so does the other. Second, the standard deviation in Equation 10-7, which we have chosen to call the standard counting error, must be... 

The significance of the standard counting error to the analytical chemist is increased because it can be used as a criterion for judging the operating conditions for" x-ray emission spectrography. 

Inasmuch as sc results from fluctuations that cannot be eliminated so long as quanta are counted, this standard deviation is the irreducible minimum for x-ray emission spectrography under ideal conditions. Not only is it a minimum, but it is also a predictable minimum. When the standard deviation, s, significantly exceeds the standard counting error, sc, it is likely that errors resembling those the analytical chemist usually encounters are superimposed upon the random fluctuations associated with the emission process. 

A comparison of standard deviation s to standard counting error sc is thus a useful criterion for the reliability of analytical results obtained by x-ray emission. To illustrate the simplest possible application of this criterion, consider again the x-ray data plotted in Figure 10-3, which are given in Table 10-1. The individual N s summarized in the table would, in x-ray emission spectrography, appear eventually as analytical results that is, as the Us of Figure 10-1 with x as their mean. For these 393 individual ATs, the standard deviation is... 

Inspection of Table 10-2 shows that the distribution is far from (1 aussian. Calculation of the standard deviation and standard counting error gives... 

The Standard Counting Error in More Complex Cases... 

If sc is to be compared with s for actual analytical results (Equation 10-2), then, of course, the sc must be properly calculated for the case in hand. In general, sc becomes larger and more complex if several kinds of counts are required to establish the analytical result. The standard counting errors for these different kinds of counts combine according to well-known rules. Several examples follow. 

The analytical chemist often wishes to express the counting error relative to the amount present of the element sought in a simple case we have... 

The capital N s obviously represent fractional standard counting errors, and the subscripts S and U denote standard and unknown. 

The last column, calculated according to Equation 10-15, is of particular interest because it relates the standard counting error to the amount present of the element sought (on the simplifying assumption, of course,... 

It is well to remember that the equations of this section deal with cases that differ fundamentally from that of Equation 10-3. This equation deals with the different errors in the result of a single measurement (that is, N or x), and the others are combinations of standard counting errors of different quantities that go to make up a complex datum that usually cannot be obtained in a single measurement. 

One important case deserves special mention. In some spectrographs, notably the Philips Autrometer (9.7), the comparison of standard with unknown is done as follows. The time At required for a preset number of counts to be given by the standard is established. The unknown is then counted for the same interval. Time is measured with such high precision that this measurement does not contribute to the over-all error. But At for the standard is subject to the fluctuation defined by Equation 10-4. The result of the comparison is therefore subject to the counting error of Equation 10-14 if no background correction is made, or to a similar counting error that is modified to allow for the background correction. 

Because it is desirable to see whether the calculated results in Table 10-3 are realistic, measurements were accordingly made in the authors laboratory on a General Electric XRD-5 D/S spectrograph (9.5) under conditions that minimized all errors except the counting error. Instead of comparing standard and unknown, as in Table 10-1, analytical lines for two elements, cobalt and iron, were compared because this could be done on a single sample. 

The results in Table 10-4 are satisfactory evidence that useful conclusions can be drawn from the standard counting error (10.4) in x-ray emission spectrography. 

B. Comparison of Standard Counting Error and Standard Deviation ... 

Up to this point, we have dealt with known individual errors and with the standard counting error sc as an operating criterion. This approach... 

Table 10-5. Analysis of Variance. Drift and Counting Error...

The last column of Table 10-5 shows (1) that the long-term source of variation clearly overshadows the short-term, the ratio of variances exceeding 70 and (2) that the short-term variance is comparable with Arx, which is, of course, the standard counting error squared (Equation 10-8). 

The first group of experiments showed that no significant drift was present the standard deviation (s = 468 counts) and 1he standard counting error (sc — 491 counts) were virtually identical. An analysis of variance for the second and third groups of experiments is summarized in Table 10-6. 

The last column of Table 10-6 shows the combined reset errors to be comparable with the counting error. (The subtraction of variances is justified because long-term drift had been proved absent in the first... 

A significant difference (above that expected from the counting error) between the up and down errors would point to backlash in the adjustment drum. Although there is a difference, as Table 10-6 shows, the E-test16 shows that this difference has a low level of significance. 

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